Algorithms for GIS:! Quadtrees

Transcription

1 Algorithms for GIS: Quadtrees

2 Quadtree A data structure that corresponds to a hierarchical subdivision of the plane Start with a square (containing inside input data) Divide into 4 equal squares (quadrants) Continue subdividing each quadrant recursively Subdivide a square until it satisfies a stopping condition: a quadrant is small enough, for e.g. contains at most 1 point

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9 Quadtrees Simple and versatile data structure Lots of applications Quadtree can be built for points edges polygons images Generalizes to d dimensions d=3: octree Many variants of quadtrees have been proposed Hundreds of papers

10 Point-quadtree Let P = set of n points in the plane Input: P Problem: Store P in a quadtree such that every square has <= 1 point Questions: 1. Size? 2. How to build it and how fast? 3. What can we do with it?

11 Exercises Let P = set of n points in the plane Draw the quadtree corresponding to a regular grid how many nodes does it have? how many leaves? height? Pick a set of points with a non-uniform distribution and draw the quadtree how many nodes does it have? how many leaves? height?

12 Exercises Let n=2 ==> we ll look at sets of 2 points in the plane. Sketch the smallest possible quad tree for two points in the plane. Sketch the largest possible quad tree for two points in the plane. Give an upper bound for the height of a quadtree for 2 points.

13 Size Let P = set of n points in the plane Theorem: The height of a quadtree storing P is at most lg (s/d) + 3/2, where s is the side of the original square and d is the distance between the closest pair of points in P. Proof: Each level divides the side of the quadrant into two. After i levels, the side of the quadrant is s/2^i. This means that The distance between points can be arbitrarily small, so the height of a quad tree can be arbitrarily large in the worst case.

14 Building a quad tree Let P = set of n points in the plane Node buildquadtree(set of points P, square S) if P has at most one point: build a leaf node, store P in it, and return node else partition S into 4 quadrants S1, S2, S3, S4 partition P into P1, P2, P3, P4 create a node node ->child1 = buildquadtree(p1, S1) node ->child2 = buildquadtree(p2, S2) node ->child3 = buildquadtree(p3, S3) node ->child4 = buildquadtree(p4, S4) return node

15 Building a quadtree Let P = set of n points in the plane How long does it take? Let the height of the quadtree be h. Analysis: Total time = total time in partitioning + total time in recursion Partitioning Partitioning P into P1, P2, P3, P4 runs in time O( P ). We cannot bound precisely each P1, P2, P3, P4 (each can have anywhere between 0 points and n points); But if we look at all nodes at same level in the quadtree: together they partition the input square and all their sets add up to precisely n The time to partition, summed over the entire quadtree, will be O(n) per level, or O(h x n) in total Recursion: Every recursive call creates a node How many nodes?

16 Building a quad tree Let P = set of n points in the plane Question: What is the total size (number of nodes) in a quadtree if height h, storing n points? Quadtree consists of leaves and internal nodes Let L_i = number of leaves Let N_i = number of internal nodes Counting leaves: each node has 0 or 4 children. Claim: The number of leaves is N_i Counting the internal nodes: Each internal node has at least two point inside it (otherwise it would satisfy the stopping criterion and would be a leaf) At each level of the quadtree: the internal nodes define a partition of the input square ==> O(n) nodes per level Total N_i = h x O(n) Total: L_i+N_i = (1+3N_i) + N_i = O(N_i) = O(h x n)

17 Building a quad tree Let P = set of n points in the plane Theorem: A quadtree for P can be built in O(hxn) time, where h is the depth (height) of the quad tree.

18 Building a quad tree Let P = set of n points in the plane Theorem: A quadtree for P can be built in O(hxn) time, where h is the depth (height) of the quad tree.

19 Point quadtree: Summary Let P = set of n points in the plane A quadtree for P has height O(lg (1/d)). A point quadtree of n points can be built in O(h x n) time. Theoretical worst case: height is unbounded in the worst case In practice: often h=o(n) and build time is O(n 2 ) Note: For sets of points that are approximately uniformly distributed, we have that h = O(lg n) and the running time becomes O( n lg n). In many practical situations quad trees have logarithmic height and can be built in O(n lg n) time

20 Point quadtree: Summary Let P = set of n points in the plane A quadtree for P has height O(lg (1/d)). A point quadtree of n points can be built in O(h x n) time. Extensions: compressed quadtrees: height is h=o(n) in the worst-case idea: compress paths of nodes with 3 empty children into one node, called a donut ==> a node may have 5 children, an empty donut + 4 regular quadrants

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23 Applications of quadtrees Hundreds of papers.. Specialized quadtrees for storing edges (edge quad trees), polygons, etc Used to answer queries on spatial data such as: find the nearest neighbor (NN) of this point find the k-nns of this point find all points in this rectangle (range searching) find all segments intersecting a given segment etc

24 Applications of quadtrees Used for fast rendering (LOD) Level i in the qdt > scene at a certain resolution bottom level has full resolution render scene at a resolution dependent on its distance from the viewpoint

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26 Applications of quadtrees Image analysis/compression